25,982 research outputs found
Transport coefficients of graphene: Interplay of impurity scattering, Coulomb interaction, and optical phonons
We study the electric and thermal transport of the Dirac carriers in
monolayer graphene using the Boltzmann-equation approach. Motivated by recent
thermopower measurements [F. Ghahari, H.-Y.~Xie, T. Taniguchi, K. Watanabe,
M.~S.~Foster, and P.~Kim, Phys.\ Rev.\ Lett.\ {\bf 116}, 136802 (2016)], we
consider the effects of quenched disorder, Coulomb interactions, and
electron--optical-phonon scattering. Via an unbiased numerical solution to the
Boltzmann equation we calculate the electrical conductivity, thermopower, and
electronic component of the thermal conductivity, and discuss the validity of
Mott's formula and of the Wiedemann-Franz law. An analytical solution for the
disorder-only case shows that screened Coulomb impurity scattering, although
elastic, violates the Wiedemann-Franz law even at low temperature. For the
combination of carrier-carrier Coulomb and short-ranged impurity scattering, we
observe the crossover from the interaction-limited (hydrodynamic) regime to the
disorder-limited (Fermi-liquid) regime. In the former, the thermopower and the
thermal conductivity follow the results anticipated by the relativistic
hydrodynamic theory. On the other hand, we find that optical phonons become
nonnegligible at relatively low temperatures and that the induced electron
thermopower violates Mott's formula. Combining all of these scattering
mechanisms, we obtain the thermopower that quantitatively coincides with the
experimental data.Comment: 20 pages, 9 figure
Unconditional Uniqueness of the cubic Gross-Pitaevskii Hierarchy with Low Regularity
In this paper, we establish the unconditional uniqueness of solutions to the
cubic Gross-Pitaevskii hierarchy on in a low regularity Sobolev
type space. More precisely, we reduce the regularity down to the currently
known regularity requirement for unconditional uniqueness of solutions to the
cubic nonlinear Schr\"odinger equation ( if and
if ). In such a way, we extend the recent work of
Chen-Hainzl-Pavlovi\'c-Seiringer.Comment: 26 pages, 1 figur
Mathematical Modeling of Product Rating: Sufficiency, Misbehavior and Aggregation Rules
Many web services like eBay, Tripadvisor, Epinions, etc, provide historical
product ratings so that users can evaluate the quality of products. Product
ratings are important since they affect how well a product will be adopted by
the market. The challenge is that we only have {\em "partial information"} on
these ratings: Each user provides ratings to only a "{\em small subset of
products}". Under this partial information setting, we explore a number of
fundamental questions: What is the "{\em minimum number of ratings}" a product
needs so one can make a reliable evaluation of its quality? How users' {\em
misbehavior} (such as {\em cheating}) in product rating may affect the
evaluation result? To answer these questions, we present a formal mathematical
model of product evaluation based on partial information. We derive theoretical
bounds on the minimum number of ratings needed to produce a reliable indicator
of a product's quality. We also extend our model to accommodate users'
misbehavior in product rating. We carry out experiments using both synthetic
and real-world data (from TripAdvisor, Amazon and eBay) to validate our model,
and also show that using the "majority rating rule" to aggregate product
ratings, it produces more reliable and robust product evaluation results than
the "average rating rule".Comment: 33 page
Stochastic phenotype transition of a single cell in an intermediate region of gene-state switching
Multiple phenotypic states often arise in a single cell with different
gene-expression states that undergo transcription regulation with positive
feedback. Recent experiments have shown that at least in E. coli, the gene
state switching can be neither extremely slow nor exceedingly rapid as many
previous theoretical treatments assumed. Rather it is in the intermediate
region which is difficult to handle mathematically.Under this condition, from a
full chemical-master-equation description we derive a model in which the
protein copy-number, for a given gene state, follow a deterministic mean-field
description while the protein synthesis rates fluctuate due to stochastic
gene-state switching. The simplified kinetics yields a nonequilibrium landscape
function, which, similar to the energy function for equilibrium fluctuation,
provides the leading orders of fluctuations around each phenotypic state, as
well as the transition rates between the two phenotypic states. This rate
formula is analogous to Kramers theory for chemical reactions. The resulting
behaviors are significantly different from the two limiting cases studied
previously.Comment: 6 pages,4 figure
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